This chapter describes some basic concepts which you have seen in earlier grades and lays the foundation for the remainder of this book. You should feel confident with the content in this chapter, before moving on with the rest of the book.

You can try out your skills on exercises in this chapter and ask your teacher for more questions just like them. You can also try to make up your own questions, solve them and try them out on your classmates to see if you get the same answers.

Practice is the only way to get good at maths!

A number is a way to represent quantity. Numbers are not something that you can touch or hold, because they are not physical. But you can touch three apples, three pencils, three books. You can never just touch three, you can only touch three of something. However, you do not need to see three apples in front of you to know that if you take one apple away, there will be two apples left. You can just think about it. That is your brain representing the apples in numbers and then performing arithmetic on them.

A number represents quantity because we can look at the world around us and quantify it using numbers. How many minutes? How many kilometers? How many apples? How much money? How much medicine? These are all questions which can only be answered using numbers to tell us “how much” of something we want to measure.

A number can be written in many different ways and it is always best to choose the most appropriate way of writing the number. For example, “a half” may be spoken aloud or written in words, but that makes mathematics very difficult and also means that only people who speak the same language as you can understand what you mean. A better way of writing “a half” is as a fraction 1212 or as a decimal number 0,50,5. It is still the same number, no matter which way you write it.

In high school, all the numbers which you will see are called real numbers and mathematicians use the symbol RR to represent the set of all real numbers, which simply means all of the real numbers. Some of these real numbers can be written in ways that others cannot. Different types of numbers are described in detail in Section 1.12.

A set is a group of objects with a well-defined criterion for membership. For example, the criterion for belonging to a set of apples, is that the object must be an apple. The set of apples can then be divided into red apples and green apples, but they are all still apples. All the red apples form another set which is a sub-set of the set of apples. A sub-set is part of a set. All the green apples form another sub-set.

Now we come to the idea of a union, which is used to combine things. The symbol for union is ∪∪. Here, we use it to combine two or more intervals. For example, if xx is a real number such that 1<x≤31<x≤3 or 6≤x<106≤x<10

where the ∪∪ sign means the union (or combination) of the two intervals. We use the set and interval notation and the symbols described because it is easier than having to write everything out in words.

The simplest thing that can be done with numbers is adding, subtracting, multiplying or dividing them. When two numbers are added, subtracted, multiplied or divided, you are performing arithmetic1. These four basic operations can be performed on any two real numbers.

Mathematics as a language uses special notation to write things down. So instead of:

mathematicians write

In earlier grades, place holders were used to indicate missing numbers in an equation.

However, place holders only work well for simple equations. For more advanced mathematical workings, letters are usually used to represent numbers.

These letters are referred to as variables, since they can take on any value depending on what is required. For example, x=1x=1 in Equation 5, but x=26x=26

A constant has a fixed value. The number 1 is a constant. The speed of light in a vacuum is also a constant which has been defined to be exactly 299 792 458 m··s-1-1(read metres per second). The speed of light is a big number and it takes up space to always write down the entire number. Therefore, letters are also used to represent some constants. In the case of the speed of light, it is accepted that the letter cc represents the speed of light. Such constants represented by letters occur most often in physics and chemistry.

Additionally, letters can be used to describe a situation mathematically. For example, the following equation

can be used to describe the situation of finding how much change can be expected for buying an item. In this equation, yy represents the price of the item you are buying, xx represents the amount of change you should get back and zz is the amount of money given to the cashier. So, if the price is R10 and you gave the cashier R15, then write R15 instead of zz and R10 instead of yy and the change is then xx.

We will learn how to “solve” this equation towards the end of this chapter.

Addition (++) and subtraction (--) are the most basic operations between numbers but they are very closely related to each other. You can think of subtracting as being the opposite of adding since adding a number and then subtracting the same number will not change what you started with. For example, if we start with aa and add bb, then subtract bb, we will just get back to aa again:

If we look at a number line, then addition means that we move to the right and subtraction means that we move to the left.

The order in which numbers are added does not matter, but the order in which numbers are subtracted does matter. This means that:

The sign ≠≠ means “is not equal to”. For example, 2+3=52+3=5 and 3+2=53+2=5, but 5-3=25-3=2 and 3-5=-23-5=-2. -2-2 is a negative number, which is explained in detail in "Negative Numbers".

Just like addition and subtraction, multiplication (××, ··) and division (÷÷, /) are opposites of each other. Multiplying by a number and then dividing by the same number gets us back to the start again:

Sometimes you will see a multiplication of letters as a dot or without any symbol. Don't worry, its exactly the same thing. Mathematicians are efficient and like to write things in the shortest, neatest way possible.

It is usually neater to write known numbers to the left, and letters to the right. So although 4x4x and x4x4 are the same thing, it looks better to write 4x4x. In this case, the “4” is a constant that is referred to as the coefficient of xx.

Brackets2 in mathematics are used to show the order in which you must do things. This is important as you can get different answers depending on the order in which you do things. For example:

whereas

If there are no brackets, you should always do multiplications and divisions first and then additions and subtractions3. You can always put your own brackets into equations using this rule to make things easier for yourself, for example:

If you see a multiplication outside a bracket like this

then it means you have to multiply each part inside the bracket by the number outside

unless you can simplify everything inside the bracket into a single term. In fact, in the above example, it would have been smarter to have done this

It can happen with letters too

If there are two brackets multiplied by each other, then you can do it one step at a time:

Now that we have described the basic rules of negative and positive numbers and what to do when you add, subtract, multiply and divide them, we are ready to tackle some real mathematics problems!

Earlier in this chapter, we wrote a general equation for calculating how much change (xx) we can expect if we know how much an item costs (yy) and how much we have given the cashier (zz). The equation is:

So, if the price is R10 and you gave the cashier R15, then write R15 instead of zz and R10 instead of yy.

Now that we have written this equation down, how exactly do we go about finding what the change is? In mathematical terms, this is known as solving an equation for an unknown (xx in this case). We want to re-arrange the terms in the equation, so that only xx is on the left hand side of the == sign and everything else is on the right.

The most important thing to remember is that an equation is like a set of weighing scales. In order to keep the scales balanced, whatever is done to one side must be done to the other.

Figure 2: An equation is like a set of weighing scales. In order to keep the scales balanced, you must do the same thing to both sides. So, if you add, subtract, multiply or divide the one side, you must add, subtract, multiply or divide the other side too.

A fraction is one number divided by another number. There are several ways to write a number divided by another one, such as a÷ba÷b, a/ba/b and abab. The first way of writing a fraction is very hard to work with, so we will use only the other two. We call the number on the top (left) the numerator and the number on the bottom (right) the denominator. For example, in the fraction 1/51/5 or 1515, the numerator is 1 and the denominator is 5.

The reciprocal of a fraction is the fraction turned upside down, in other words the numerator becomes the denominator and the denominator becomes the numerator. So, the reciprocal of 2323 is 3232.

A fraction multiplied by its reciprocal is always equal to 1 and can be written

This is because dividing by a number is the same as multiplying by its reciprocal.

A decimal number is a number which has an integer part and a fractional part. The integer and the fractional parts are separated by a decimal point, which is written as a comma in South African schools. For example the number 314100314100 can be written much more neatly as 3,143,14.

All real numbers can be written as a decimal number. However, some numbers would take a huge amount of paper (and ink) to write out in full! Some decimal numbers will have a number which will repeat itself, such as 0,33333...0,33333... where there are an infinite number of 3's. We can write this decimal value by using a dot above the repeating number, so 0,3˙=0,33333...0,3˙=0,33333.... If there are two repeating numbers such as 0,121212...0,121212... then you can place dots4 on each of the repeated numbers 0,1˙2˙=0,121212...0,1˙2˙=0,121212.... These kinds of repeating decimals are called recurring decimals.

Table 5 lists some common fractions and their decimal forms.

In science one often needs to work with very large or very small numbers. These can be written more easily in scientific notation, which has the general form

where aa is a decimal number between 0 and 10 that is rounded off to a few decimal places. The mm is an integer and if it is positive it represents how many zeros should appear to the right of aa. If mm is negative, then it represents how many times the decimal place in aa should be moved to the left. For example 3,2×1033,2×103 represents 32 000 and 3,2×10-33,2×10-3 represents 0,00320,0032.

If a number must be converted into scientific notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 (i.e. we need to work out the value of the exponent mm) and what this number is (the value of aa). We do this by counting the number of decimal places the decimal point must move.

For example, write the speed of light which is 299 792 458 ms-1-1 in scientific notation, to two decimal places. First, determine where the decimal point must go for two decimal places (to find aa) and then count how many places there are after the decimal point to determine mm.

In this example, the decimal point must go after the first 2, but since the number after the 9 is a 7, a=3,00a=3,00.

So the number is 3,00×10m3,00×10m, where m=8m=8, because there are 8 digits left after the decimal point. So, the speed of light in scientific notation to two decimal places is 3,00×1083,00×108 ms-1-1.

As another example, the size of the HI virus is around 1,2×10-71,2×10-7 m. This is equal to 1,2×0,00000011,2×0,0000001 m, which is 0,00000012 m.

Now that we have learnt about the basics of mathematics, we can look at what real numbers are in a little more detail. The following are examples of real numbers and it is seen that each number is written in a different way.

Depending on how the real number is written, it can be further labelled as either rational, irrational, integer or natural. A set diagram of the different number types is shown in Figure 3.

Figure 3: Set diagram of all the real numbers RR, the rational numbers QQ, the integers ZZ and the natural numbers NN. The irrational numbers are the numbers not inside the set of rational numbers. All of the integers are also rational numbers, but not all rational numbers are integers.

Rational Numbers

The natural numbers and the integers are only able to describe quantities that are whole or complete. For example, you can have 4 apples, but what happens when you divide one apple into 4 equal pieces and share it among your friends? Then it is not a whole apple anymore and a different type of number is needed to describe the apples. This type of number is known as a rational number.

A rational number is any number which can be written as:

a b a b

(36)

where aa and bb are integers and b≠0b≠0.

The following are examples of rational numbers:

20 9 , - 1 2 , 20 10 , 3 15 20 9 , - 1 2 , 20 10 , 3 15

(37)

Notation Tip

Rational numbers are any number that can be expressed in the form ab;a,b∈Z;b≠0ab;a,b∈Z;b≠0 which means “the set of numbers abab when aa and bb are integers”.

Mathematicians use the symbol QQ to mean the set of all rational numbers. The set of rational numbers contains all numbers which can be written as terminating or repeating decimals.

Rational Numbers

All integers are rational numbers with a denominator of 1.

You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integers a,b,ca,b,c and dd, then the rules for adding and multiplying rational numbers are

a b + c d = a d + b c b d a b × c d = a c b d a b + c d = a d + b c b d a b × c d = a c b d

(38)

Notation Tip

The statement "4 integers a,b,ca,b,c and dd" can be written formally as {a,b,c,d}∈Z{a,b,c,d}∈Z because the ∈∈ symbol means in and we say that a,b,ca,b,c and dd are in the set of integers.

Two rational numbers (abab and cdcd) represent the same number if ad=bcad=bc. It is always best to simplify any rational number, so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. For example, the rational number 1000/100001000/10000 can be divided by 1000 on the top and the bottom, which gives 1/101/10. 2323 of a pizza is the same as 812812 (Figure 4).

Figure 4: 812812 of the pizza is the same as 2323 of the pizza.

You can also add rational numbers together by finding the lowest common denominator and then adding the numerators. Finding a lowest common denominator means finding the lowest number that both denominators are a factor5 of. A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3

3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , ... 3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , ...

(39)

and the following all have factors of 4

4 , 8 , 12 , 16 , 20 , 24 , 28 , ... 4 , 8 , 12 , 16 , 20 , 24 , 28 , ...

(40)

The common denominators between 3 and 4 are all the numbers that appear in both of these lists, like 12 and 24. The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is 12.

For example, if we wish to add 34+2334+23, we first need to write both fractions so that their denominators are the same by finding the lowest common denominator, which we know is 12. We can do this by multiplying 3434 by 3333 and 2323 by 4444. 3333 and 4444 are really just complicated ways of writing 1. Multiplying a number by 1 doesn't change the number.

3 4 + 2 3 = 3 4 × 3 3 + 2 3 × 4 4 = 3 × 3 4 × 3 + 2 × 4 3 × 4 = 9 12 + 8 12 = 9 + 8 12 = 17 12 3 4 + 2 3 = 3 4 × 3 3 + 2 3 × 4 4 = 3 × 3 4 × 3 + 2 × 4 3 × 4 = 9 12 + 8 12 = 9 + 8 12 = 17 12

(41)

Dividing by a rational number is the same as multiplying by its reciprocal, as long as neither the numerator nor the denominator is zero:

a b ÷ c d = a b . d c = a d b c a b ÷ c d = a b . d c = a d b c

(42)

A rational number may be a proper or improper fraction.

Proper fractions have a numerator that is smaller than the denominator. For example,

- 1 2 , 3 15 , - 5 - 20 - 1 2 , 3 15 , - 5 - 20

(43)

are proper fractions.

Improper fractions have a numerator that is larger than the denominator. For example,

- 10 2 , 15 13 , - 53 - 20 - 10 2 , 15 13 , - 53 - 20

(44)

are improper fractions. Improper fractions can always be written as the sum of an integer and a proper fraction.

Converting Rationals into Decimal Numbers

Converting rationals into decimal numbers is very easy.

If you use a calculator, you can simply divide the numerator by the denominator.

If you do not have a calculator, then you have to use long division.

Since long division was first taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you.

The following is a table of the meanings of some mathematical signs and symbols that you should have come across in earlier grades.

So if we write x>5x>5, we say that xx is greater than 5 and if we write x≥yx≥y, we mean that xx can be greater than or equal to yy. Similarly, << means `is less than' and ≤≤ means `is less than or equal to'. Instead of saying that xx is between 6 and 10, we often write 6<x<106<x<10. This directly means `six is less than xx

Infinity (symbol ∞∞) is usually thought of as something like “the largest possible number" or “the furthest possible distance". In mathematics, infinity is often treated as if it were a number, but it is clearly a very different type of “number" than integers or reals.

When talking about recurring decimals and irrational numbers, the term infinite was used to describe never-ending digits.